Furthermore, we denote \(\mu _t\)  the empirical distribution of \(\lbrace \theta _{\tau \in [t]}\rbrace \)  at time \(t\) , that is \(\mu _t(\theta ):=\sum _{\tau \in [t]}\mathbb {I}(\theta _t=\theta )/t\) , where \(\mathbb {I}(\cdot )\)  is the indicator function. Then, it can be seen that \(\sigma _{t+1}\)  is the solution to (REF ) with \(\mu ^*\)  being replaced by \(\mu _t\) . In other words, the leader always chooses the signaling commitment specified by the sigBSE, with the prior being \(\mu _t\) . According to the concentration bound, \(\mu _t\)  converges to \(\mu ^*\)  in point-wise with high probability (i.i.d. sampling) [1]}, and naturally, (REF ) gives a desired online performance. We state our result formally in the following theorem.
